1251 
of two &*, of rank 36. As 6? is of rank four and 0° as rational 
curve of rank eight, while s has four points in common with @?, 
gv and o° will have eight points in common. We can therefore 
determine the congruence [v°| as the system of rational curves 9° 
passing through three fundamental points FF, F,, cutting the singular 
curve 0° eight times and having s as singular quadrisecant. 
It incidentally follows from this, that g° may satisfy 20 simple 
conditions. 
2. Let b be a bisecant of 6°, resting on s; all ® passing through 
a point of 4 have this line in common, theretore determine a pencil 
the base of which consists of s, 6, o* and a rational 0*, which has 
three points with s six points with 0°, consequently one point in 
common with 6. 
There are also figures of [g°| consisting of a conic g° and a 
cubic 9’. The plane ®, passing through /’, and s forms with the 
ruled surface ®,’, determined by 6%, #, and /,, a ®,*. Any other 
figure of [®°] intersects ®, along a conic ge,’ in the plane ®,, 
passing through #, and the intersections S,“ of 60°, and a twisted 
curve uv," intersecting 6* in five points C,, C,’, which are determined 
by ®,°; it passes of course through #, and 4. 
To the curves 0,* belong two degenerate figures each formed by 
the bisecant of 6° out of one of the points C, and the conic ,*, in 
which ®,° is intersected by the plane that connects the points /’, 
and /’, with the other point C. Apparently p‚* and the corresponding 
a 
9, form a degenerate curve g°. 
N 
The three degenerate conics g,* as well determine degenerate 
curves o*. For the straight line S,'S," is a bisecant 6; hence the 
line FS," forms with the corresponding o,* a degenerate figure ¢*. 
3. To the net [®*] belongs the surface >*, which has a node 
in a point S of o’. This nodal surface determines with any other 
surface of the net a 9°, intersecting o* in S, is therefore the locus 
of the g° passing through the singular point S. 
The surfaces ,* and =,* have s, o* and a 9° in common, conse- 
quently one @° passes through two points S,, S, of o°. The groups 
of eight points, which the curves of the congruence determine on 
6° form therefore an involution of the second rank. From this ensues 
that o*® is osculated by 18 curves 0°, and contains 21 pairs S,, 5, 
through which oe curves g° pass. So there are 21 surfaces ®*° each 
possessing fwo nodes lying on 6°, 
A straight line passing through the vertex S of the monotd =* inter- 
83 
Proceedings Royal Acad. Amsterdam. Vol, XVII. 
